Does $f^{-1}(f(S))=S$ for $S\subseteq A$ imply that $f|_S$ is a bijection from $S$ to $f(S)$? Let $f:A\to B$ be a function.  Does $f^{-1}(f(S))=S$ for $S\subseteq A$ imply that $f|_S$ is a bijection from $S$ to $f(S)$?
My thinking was that $f^{-1}$ exhibits the property of an inverse function going from $f(S)$ to $S$, so therefore $f(S)\subseteq B$ and $S\subseteq A$ are in bijective correspondence, or in other words $f|_S$ is a bijection.
Is this correct?
 A: Notice that in the expression $f^{-1}(f(S))=S$, the object on which one is applying the symbol $f^{-1}$, namely the object $f(S)$, is not an element of the set $B$. Instead, $f(S)$ is a subset of the set $B$. So in that expression, the symbol $f^{-1}$ should be regarded as a function $f^{-1} : \mathcal P(B) \to \mathcal P(A)$, whose domain is the power set $\mathcal P(B)$ of $B$ and whose range is the power set $\mathcal P(A)$ of the set $A$: you input a subset of $B$ to $f^{-1}$ and the output is a subset of $A$. This can always be done for any function $f : A \to B$, regardless of whether $f$ is injective or surjective or anything else. In your specific situation, given a subset $S \subset A$ whose image under $f$ is the subset $f(S) \subset B$, you input $f(S)$ to $f^{-1}$ and the output is $f^{-1}(f(S)) \subset A$.
So no, your idea for a proof is not correct. One can indeed sometimes prove that $f : A \to B$ is a bijection by producing an appropriate function $B \to A$. But producing a function $\mathcal P(B) \to \mathcal P(A)$ does not seem to me to be a viable strategy for proving that $f$ is a bijection.
You might wish that the symbol $f^{-1}$ was not used in such a misleading and confusing way, but that's how it goes, this is a very common usage in set theory.
As for your question itself, I'm quite confident that you can find a counterexample, i.e. a function $f : A \to B$ and a subset $S \subset A$ such that $f^{-1}(f(S))=S$ and yet $f\!\mid_S : S \to f(S)$ is not a bijection. I suggest starting your search for a counterexample using sets $A,B$ of the very smallest cardinalities.
